Samer Adeeb

Approximate Methods: Point Collocation Method

The statement of the equilibrium equations applied to a set D\subset\mathbb{R}^3 is as follows. Assuming that at equilibrium \sigma:D\rightarrow\mathbb{M}^3 is the symmetric Cauchy stress distribution on D and that u:D\rightarrow\mathbb{R}^3 is the displacement vector distribution and knowing the relationship \sigma=\sigma(u), then the equilibrium equation seeks to find u\in U such that the associated \sigma satisfies the equation:

    \[ \mathrm{div}\sigma+\rho b= 0 \]

where b:D\rightarrow\mathbb{R}^3 is the body forces vector distribution on D, \rho is the mass density, and U is the space of all possible displacement functions applied to D, i.e., U=\{a:D\rightarrow\mathbb{R}^3| a\mbox{ is kinematically admissible}\}. The term “Kinematically admissible” in U indicates that the space of possible solutions must satisfy the boundary conditions imposed on \partial D_u (as stated below) and any differentiability constraints.

The boundary conditions for the equations of equilibrium are usually given on two parts of the boundary of D denoted \partial D. On the first part, \partial D_n, the external traction vectors t_n are known so we have the boundary conditions for \sigma since \sigma^Tn=t_n (n is the normal vector to the boundary). On the second part, \partial D_u, the displacement is given.

The point collocation method seeks an approximate solution to the above problem by assuming that the solution u has a particular form with a finite number of unknowns, i.e., by looking for u in a subset U_{fd}\subset U that is finite dimensional but still able to approximate functions in U. The finite number of unknowns can be found by satisfying the differential equation of equilibrium at a number of chosen points equal to the number of unknowns.

Example

Using a polynomial function of the third degree, find the displacement function of the shown bar by satisfying the essential boundary conditions, the differential equation of equilibrium at X_1=\frac{L}{2} and at X_1=L., and the nonessential boundary condition at X_1= L. Assume that the bar is linear elastic with Young’s modulus E and cross-sectional area A and that the small strain tensor is the appropriate measure of strain. Ignore the effect of Poisson’s ratio.
PL1

Solution
Exact Solution

The exact solution can be obtained by directly solving the differential equation of equilibrium utilizing \sigma_{11}=E\frac{\mathrm{d}u_1}{\mathrm{d}X_1}:

    \[ \frac{\mathrm{d}^2u_1}{\mathrm{d}X_1^2}=-\frac{cX_1}{EA} \]

with the boundary conditions: @X_1=0:u_1=0 and X_1=L:\frac{\mathrm{d}u_1}{\mathrm{d}X_1}=0.
Therefore:

    \[ u_1=\frac{cL^2}{2EA}X_1-\frac{c}{6EA}X_1^3 \]

View Mathematica Code
DSolve[{u''[X1] == -c*X1/EA, u'[L] == 0, u[0] == 0}, u[X1], X1]
Approximate Solution

A polynomial of the third degree approximate solution has the form:

    \[ u_{approx}=a_0+a_1X_1+a_2X_1^2+a_3X_1^3 \]

with

    \[\begin{split} \frac{\mathrm{d}u_{approx}}{\mathrm{d}X_1}&=a_1+2a_2X_1+3a_3X_1^2\\ \frac{\mathrm{d}^2u_{approx}}{\mathrm{d}X_1^2}&=2a_2+6a_3X_1 \end{split} \]

The following four equations will be utilized to find the four unknowns:

    \[\begin{split} @X_1=0&:u_{approx}=0\\ @X_1=L&:\frac{\mathrm{d}u_{approx}}{\mathrm{d}X_1}=0\\ @X_1=\frac{L}{2}&:\frac{\mathrm{d}^2u_{approx}}{\mathrm{d}X_1^2}\bigg|_{X_1=\frac{L}{2}}=-\frac{cX_1}{EA}\bigg|_{X_1=\frac{L}{2}}\\ @X_1=L&:\frac{\mathrm{d}^2u_{approx}}{\mathrm{d}X_1^2}\bigg|_{X_1=L}=-\frac{cX_1}{EA}\bigg|_{X_1=L} \end{split} \]

The approximate solution that satisfies the above four unknowns has the form:

    \[ u_{approx}=\frac{cL^2}{2EA}X_1-\frac{c}{6EA}X_1^3 \]

Notice that since the exact solution is contained in the space of approximate functions which were chosen to satisfy both the essential and nonessential boundary conditions, the solution obtained is indeed the exact solution!
View Mathematica Code

u = a0 + a1*x1 + a2*x1^2 + a3*x1^3;
ux = D[u, x1]
uxx = D[ux, x1]
Eq1 = (uxx + c*x1/EA) /. x1 -> L/2;
Eq2 = (uxx + c*x1/EA) /. x1 -> L;
Eq3 = (ux) /. x1 -> L;
Eq4 = u /. x1 -> 0;
s = Solve[{Eq1 == 0, Eq2 == 0, Eq3 == 0, Eq4 == 0}, {a1, a2, a3, a0}]
u /. s[[1]]

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